*a Euclidean horror story*

You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone… Learn from my example.

*Farkas Bolyai, 1820*

I came over the dunes a little before dawn. Already waiting for me were three Examiners: robed, masked, inhumanly tall. Their silhouettes loomed against the dark ocean; their silence loomed, too, against the dull grinding of the waves.

“I’m ready for the final test,” I told them. I wasn’t sure I wanted to pass, but I knew I didn’t want to fail.

The first Examiner spoke in a thunderous rasp: “Then your task is simple.”

The second Examiner explained: “Draw two parallel lines in the sand.”

I looked to the third Examiner, who said nothing. No further instructions came.

Was that all, then?

I grabbed a dry stick of driftwood, and scraped two parallels, a few feet in length, into the damp earth.

“Longer,” said the first Examiner.

I extended the lines.

“Longer,” said the second Examiner.

I did it again.

The third Examiner shook its head, and I began to sense the nature of the task.

It was arbitrary, a measure only of blind obedience.

(Maybe all tests are, in the end.)

I fetched a second stick, similar in length to the first. Then I walked backwards along the beach, dragging the two sticks behind me, one in each hand. The Examiners followed, seeming to glide, leaving no footprints. I extended the parallel lines until they were a quarter-mile, a half-mile, a full mile long.

“How much further?” I asked, perhaps an hour after sunrise.

No one spoke. I counted a dozen waves before the first Examiner replied: “Do you know the meaning of ‘parallel’?”

“It means the lines never meet,” I said.

The second Examiner nodded. “Prove that you know what this signifies.”

I shivered, despite the building heat of day. “Why? I know what ‘never’ is.”

“Do you?” the first Examiner asked. “You scratch symbols, but symbols of what? You mutter sounds, but are there thoughts behind them? A mortal mouth can mimic immortal speech. It does not mean you understand.”

Ah. So this was the test.

I picked up the sticks and began to scrape again. The sun continued to climb. I watched my shadow shorten, and couldn’t help noticing that the Examiners cast no shadows of their own.

By the early afternoon, I was sweating, stumbling, panting with thirst. I stopped to rest, though I knew this displeased the Examiners.

“I know what parallel lines are,” I told them again.

“Do you?” the second Examiner asked. “You watched the Archer of Time notch his first arrow? You observed the Sculptor of Space, when this universe was mere potter’s clay on a spinning wheel? You, a mortal creature, flinching at pain—you claim to know eternity?”

There was nothing to say to that.

I continued to scratch lines in the wet sand, step by backwards step, my body fraying with thirst, my worthless work trailing into the distance.

Hours later, as the sun set over the dunes, I fell to my knees. “I can’t go on,” I told them. My hoarse voice scarcely sounded like my own.

“Then you don’t understand,” concluded the first Examiner.

“Parallel lines never meet,” whispered the second. “Never, never, never.”

The third Examiner said nothing.

What, I wondered, was the purpose of this test? Did it sift the wise from the rash, the patient from the arrogant? Or did it exist only to separate test-givers from test-takers? Did it merely flaunt their power to demand—and my powerlessness to refuse?

I felt a surge of impotent anger, and lashed out the only way I could: I drew the two sticks together, crossing the lines in the sand. “There,” I said. “There are your parallel lines—at least, the closest you’ll have from my hands.”

I braced myself for their final judgment, for the fatal pressure on the back of my neck.

But when I looked up, two of the Examiners had vanished. Only the third remained. I saw its outstretched fingers, elongated and skeletal, and I sensed that it was smiling behind its mask.

“Congratulations,” the Examiner said, with a sinister calm. “You do understand.”

it’s a little like “what’s the airspeed velocity of an unladen swallow?”

You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone…Learn from my example.

–Frakas Bolyai to is son Janos

YES DOUG! It didn’t occur to me how perfect that quote is for this setting! I *love* Farkas Bolyai and that letter to Janos – I’ve used it in talks and lessons – but your excerpt totally suits the story. I’m gonna add it as an epigraph.

Does this mean that at the very end parallel lines DO meet?

it depends on your postulates.

If you’re drawing lines on Earth, then they will eventually cross twice.

Except maybe if I draw them along the equatorial plane.

The only line of latitude which is a straight line is the Equator. All other “lines” of latitude are circles.

The Examiners represent the Euclidean, hyperbolic, and projective geometries, respectively?

i bet one of them looks like debbie harry.

Growing up in England I was taught that two lines that are parallel to each other are (1) always straight, (2) always the same distance apart, and (3) meet at infinity (which essentially means they never meet).

Reading the story here it occurred to me (as a non-mathematician but fascinated by the beauty of mathematics, once I acquired an inkling of what that might mean) that two lines could be drawn in the form of, for example, two concentric circles. Neither line will ever meet the other, they’ll always be the same distance apart, but they won’t be straight. Are they still parallel?

If I deform the circles tremendously but keep one figure inside the other such that their circumferences never touch, still neither line will ever meet the other, their distance apart will vary but always be constrained, and they won’t be straight. Are they still parallel?

I no longer think I really know what parallel means…

A crash course in non-Euclidean Geometry.

In order to be a line you must satisfy the first postulate. Between any two points there exists a unique line.

Living on the a spherical planet, a straight line is the shortest distance between two points. i.e. a great circle path. But what about antipodal points? There are multiple great circle paths connecting the north and south pole. From the point of view of the geometers, they side step this, and say that antipodes are equivalent points.

In a spherical geometry, all lines intersect and none are parallel.

An important non-Euclidean geometry is projective geometry. Take “flat” 3D space and project it onto a 2D plane. If you have ever learned how to draw with prospective in an art class, that is a projective geometry. If lines were parallel in Eculidean 3-space, we still call them parallel, as we represent them on the plane. This means that parellel lines can head into a vanishing point into your drawing, intersect on the page, and still be considered parallel.

Parallel lines do exist, but they appear to intersect.

And a 3rd example is hyperbolic geometry. This is a tougher one to visualize. Suppose you have some sort of plastic that that you can stretch out, but once stretched will not return to its old shape.

You pull it into a disc, and then you keep stretching the rim until it is just a little bit to big to lie flat. The edge is just a little bit curly. You can locally flatten it out, but that just pushes the curly bits somewhere else. Now, if you start to draw a pair of parallel lines, as you extend from the starting point, and flatten the local area while you extend the lines. your lines will move farther apart as you go.

In a hyperbolic geometry, given a line and a point not on the line, there are multiple lines through that point that will be parallel to the given line.

Assuming Euclidean geometry, if they’re circles, then regardless of the definition of “parallel”, they can’t be parallel lines because they’re not lines (being straight is part of the definition of a line). (I don’t know if or how “parallel” is defined on non-lines.)

Way over my head and yet …. still enjoyable.

Once you see them accept such wavy, clearly not-straight scratches as ‘lines’, then your way out is clear. Draw two circles. The lines are as long as they can possibly be and never meet. For extra credit, consider flipping off the examiners because they were just messing with you anyway.

Nice,quiet n thoughtful